Second and Third order differential subordination for exponential function

Abstract

This article presents several findings regarding second and third-order differential subordination of the form: $$p(z)+{\hat{I}}_1^{3}z{p}^{\prime }(z)+{\hat{I}}_2^3{z}^2{p}^{\prime \prime }(z)\prec h(z)⟹p(z)\prec e^z$$ and $$p(z)+{\hat{I}}_1^{3}z{p}^{\prime }(z)+{\hat{I}}_2^3{z}^2{p}^{\prime \prime }(z)+{\hat{I}}_3^3{z}^3{p}^{\prime \prime \prime }(z)\prec h(z)⟹p(z)\prec e^z.$$ Here, ${\hat{I}}_1^3$, ${\hat{I}}_2^3$, and ${\hat{I}}_3^3$ represent positive real numbers, and various selections of $h(z)$ are explored within the context of the class ${\mathcal{S}}_e^{\ast }:=\{f\in \mathcal{A}:z{f}^{\prime }(z)/f(z)\prec e^z\}$, which denotes the class of starlike functions associated with the exponential function.

arXiv admin note: substantial text overlap with arXiv:2306.11215